Most relevant logics are paraconsistent. See the entries on relevance logic , paraconsistent logic , and dialetheism. Or see Anderson and Belnap [], Anderson, Belnap, and Dunn [], and Tennant [] for fuller overviews of relevant logic; and Priest [a,b], for dialetheism. Deep philosophical issues concerning the nature of logical consequence are involved. Far be it for an article in a philosophy encyclopedia to avoid philosophical issues, but space considerations preclude a fuller treatment of this issue here.

Suffice it to note that the inference ex falso quodlibet is sanctioned in systems of classical logic , the subject of this article. It is essential to establishing the balance between the deductive system and the semantics see §5 below.

Recall that the only closed terms in our system are constants. The introduction clause for the universal quantifier is a bit more complicated. It could be any object, and so anything we conclude about it holds for all objects.

This elimination rule also corresponds to a common inference. The introduction rule is about a simple as can be:. Suppose, for example, that Harry is identical to Donald since his mischievous parents gave him two names. Again, this clause allows proofs by induction on the rules used to establish an argument.

To illustrate the level of rigor, we begin with a lemma that if a sentence does not contain a particular closed term, we can make small changes to the set of sentences we prove it from without problems. Lemma 7. The cases for the other rules are similar.

Theorem 8. The rule of Weakening. Most of the other cases are exactly like this. Theorem 8 allows us to add on premises at will. Some systems of relevant logic do not have weakening, nor does substructural logic See the entries on relevance logic , substructural logics , and linear logic. Theorem 9. Theorem The rule of Cut. The remaining cases are similar. Theorem 11 allows us to chain together inferences. This fits the practice of establishing theorems and lemmas and then using those theorems and lemmas later, at will.

The cut principle is, some think, essential to reasoning. In some logical systems, the cut principle is a deep theorem; in others it is invalid. The system here was designed, in part, to make the proof of Theorem 11 straightforward. That is, a set is consistent if it does not entail a pair of contradictory opposite sentencess. The Lindenbaum Lemma. Notice that this proof uses a principle corresponding to the law of excluded middle.

Intuitionists, who demur from excluded middle, do not accept the Lindenbaum lemma. It is what the variables range over.

The interpretation function assigns appropriate extensions to the non-logical terms. In particular,. Thus we assume that every constant denotes something. Systems where this is not assumed are called free logics see the entry on free logic.

The role of variable-assignments is to assign denotations to the free variables of open formulas. If the language contained function symbols, the denotation function would be defined by recursion.

This is about as straightforward as it gets. This takes care of the atomic formulas. We now proceed to the compound formulas of the language, more or less following the meanings of the English counterparts of the logical terminology. The final clause is similar. Theorem 6, unique readability, assures us that this definition is coherent. At each stage in breaking down a formula, there is exactly one clause to be applied, and so we never get contradictory verdicts concerning satisfaction.

As indicated, the role of variable-assignments is to give denotations to the free variables. We now show that variable-assignments play no other role. So we define. A straightforward induction establishes the following:. The definition corresponds to the informal idea that an argument is valid if it is not possible for its premises to all be true and its conclusion false.

Our definition of logical consequence also sanctions the common thesis that a valid argument is truth-preserving — to the extent that satisfaction represents truth. Validity is the model-theoretic counterpart to deducibility. A sentence is logically true if and only if it is a consequence of the empty set. Logical truth is the model-theoretic counterpart of theoremhood. So a set of sentences is satisfiable if it has a model. Satisfiability is the model-theoretic counterpart to consistency. This is a model-theoretic counterpart to ex falso quodlibet see Theorem We have the following, as an analogue to Theorem We now present some results that relate the deductive notions to their model-theoretic counterparts.

The first one is probably the most straightforward. So one would expect that an argument is deducible, or deductively valid, only if it is semantically valid. The other cases are about as straightforward.

Corollary But this is impossible, given the clause for negation in the definition of satisfaction. For all we know so far, we may not have included enough rules of inference to deduce every valid argument. The converses to soundness and Corollary 19 are among the most important and influential results in mathematical logic. We begin with the latter. Gödel []. Proof: The proof of completeness is rather complex.

We only sketch it here. One interesting feature of this construction, due to Leon Henkin, is that we build an interpretation of the language from the language itself, using some of the constants as members of the domain of discourse. Three-place predicates, etc. are interpreted similarly.

The other cases follow from the various clauses in the definition of satisfaction. Soundness and completeness together entail that an argument is deducible if and only if it is valid, and a set of sentences is consistent if and only if it is satisfiable. So we can go back and forth between model-theoretic and proof-theoretic notions, transferring properties of one to the other.

Compactness holds in the model theory because all derivations use only a finite number of premises. The interpretation we produced was itself either finite or denumerably infinite. Thus, we have the following:. Löwenheim-Skolem Theorem. There is a stronger version of Corollary Notice that if two interpretations are equivalent, then they satisfy the same sentences. Downward Löwenheim-Skolem Theorem. Proof: Like completeness, this proof is complex, and we rest content with a sketch.

The downward Löwenheim-Skolem theorem invokes the axiom of choice, and indeed, is equivalent to the axiom of choice see the entry on the axiom of choice.

But we cannot rest content with the Skolem-hull, however. This also relies on the axiom of choice. Unfortunately, space constraints require that we leave this step as an exercise.

Another corollary to Compactness Corollary 22 is the opposite of the Löwenheim-Skolem theorem:. Upward Löwenheim-Skolem Theorem. One can interpret the other new constants at will. No satisfiable set of sentences can guarantee that its models are all denumerably infinite, nor can any satisfiable set of sentences guarantee that its models are uncountable.

See the entry on second-order and higher-order logic. Such interpretations are among those that are sometimes called unintended , or non-standard models of arithmetic. That is, any first-order, satisfiable set theory or theory of the real numbers, has unintended models the size of the natural numbers.

This is despite the fact that a sentence seemingly stating that the universe is uncountable is provable in most set-theories. Surely, logic has something to do with correct reasoning, or at least correct deductive reasoning.

The details of the connection are subtle, and controversial — see Harman [] for an influential study. It is common to say that someone has reasoned poorly if they have not reasoned logically, or that a given deductive argument is bad, and must be retracted, if it is shown to be invalid.

Some philosophers and logicians have maintained that there is a single logical system that is uniquely correct, in its role of characterizing validity. Among those, some, perhaps most, favor classical, first-order logic as uniquely correct, as the One True Logic. See, for example, Quine [], Resnik [], Rumfitt [], Williamson [], and a host of others. That classical, first-order logic should be given this role is perhaps not surprising.

It has rules which are more or less intuitive, and is simple for how strong it is. As we have seen in section 5, classical, first-order logic has interesting and important meta-theoretic properties, such as soundness and completeness, that have lead to many important mathematical and logical studies.

However, as noted, the main meta-theoretic properties of classical, first-order logic lead to expressive limitations of the formal languages and model-theoretic semantics. Key notions, like finitude, countability, minimal closure, natural number, and the like cannot be expressed.

Other criticisms of classical, first-order logic have also been lodged. There are two main options available to those who are critical of classical, first-order logic, as the One True Logic. One is to propose some other logic as the One True Logic. Priest [a] describes the methodology one might use to settle in the One True Logic. The other main option is to simply deny that there is a single logic that qualifies as the One True Logic.

One instance of this is a kind of logical nihilism , a thesis that there is no correct logic. Another is a logical pluralism , the thesis that a variety of different logical all qualify as correct, or best, or even the true logic, at least in various contexts.

Of course, this is not the place to pursue this matter in detail. See Beall and Restall [] and Shapiro [] for examples of pluralism, and the entry on logical pluralism for an overview of the terrain for both logical pluralism and logical nihilism. We close with brief sketches of some of the main alternatives to classical, first-order logic, providing references to other work and entries to this Encyclopedia. See also the second half of Shapiro and Kouri Kissel [].

In recent years, some work has been done to "approximate" classical logic. The idea is to get as close to classical logic as possible, in order to preserve some of the benefits, while at the same time removing some limitations of classical logic, like being closer to intuitive inference or applying to things like vagueness and paradoxes.

For example, Barrio, Pailos and Szmuc [] show that we can approximate classical logic in something called the ST-hierarchy ST for strict-tolerant, from Cobreros, Egre, Ripley and van Rooij [a,b]. This allows them to avoid certain classical problems at each level of the hierarchy, like some of the paradoxes, while at the same time maintaining many of the benefits of the strength of classical logic when considering the full hierarchy.

Notably, he claims it solves at least the Sorites and Liar Paradoxes see the entries on the sorites paradox and liar Paradox. The system conservatively extends classical logic. Ripley claims that this is what makes it classical. However, the system is not transitive, and does not have a Cut rule. There are, of course, some questions about whether these new logics are really classical, but it is informative work nonetheless. One way to extend classical, first-order logic is to add additional operators to the underlying formal language.

Modal logic adds operators which designate necessity and possibility. So, we can say that a proposition is possibly true, or necessarily true, rather than just true. O Quine [] once argued that it is not coherent for quantifiers to bind variables inside modal operators, but opinion on this matter has since changed considerably see, for example, Barcan [].

There is now a thriving industry of developing modal logics to capture various kinds of modality and temporal operators. See the entry on modal logic. All of the formal languages sketched above have only one sort of variable. These are sometimes called first-order variables. Each interpretation of the language has a domain, which is the range of these first-order variables. It is what the language is about, according to the given interpretation.

Second-order variables range over properties, sets, classes, relations, or functions of the items in that domain. Third-order variables range over properties, classes, relations of whatever is in the range of the second-order variables.

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